Thermal Modeling of a Vibrational Mill - Industrial & Engineering

Accordingly, this paper is focused on the thermal analysis of a vibrational mill by means of a mathematical model representing the mill as a sum of pa...
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Ind. Eng. Chem. Res. 2003, 42, 2015-2021

2015

Thermal Modeling of a Vibrational Mill J. M. Castillo,†,‡ R. Lapasin,‡ and M. Grassi*,† Research Department, Eurand International SpA, via del Follatoio 12, I-34148 Trieste, Italy, and Department of Chemical, Environmental and Raw Materials Engineering, DICAMP, Piazzale Europa 1, I-34127, Trieste, Italy

A cogrinding process represents a possible strategy to improve the bioavailability of poorly watersoluble drugs because it allows the formation of composite materials (carrier plus drug) where the drug appears in the form of nanocrystals or in an amorphous phase which competes higher water solubility. Because of the fact that many drugs are thermolabile, an accurate knowledge of thermal conditions experienced by a drug during cogrinding is of paramount importance. Accordingly, this paper is focused on the thermal analysis of a vibrational mill by means of a mathematical model representing the mill as a sum of parts where temperature is spatially uniform but changing with time. The only exception is given by the mill bowl where, because of the low heat conductivity and high thickness of its wall, a two-dimensional heat conduction problem is solved. The comparison between the model best fitting and the experimental evidences confirms the reliability of the hypotheses on which the model was built. 1. Introduction Many pharmaceutical systems are essentially made up of a cross-linked polymeric carrier hosting the active agent (drug) inside its three-dimensional network,1,2 because this technique can represent a profitable tool to modulate drug-release kinetics. In particular, an increase of the dissolution rate in aqueous media and, thus, an improvement of the bioavailability (defined as the rate and extent to which the active drug is absorbed from a pharmaceutical form and becomes available at the site of drug action3) of slightly water-soluble crystalline drugs can be achieved.4 Indeed, by means of different techniques (solvent swelling,4,5 supercritical carbon dioxide,6,7 and cogrinding8,9), it is possible to disperse the drug inside the polymeric network in the form of very small crystals ranging from the nanoscale to the virtually zero-dimensional crystals represented by drug-free molecules (this corresponds to the drug amorphous state).4,8,9 The interesting aspect of drug nanocrystals and the amorphous state relies on the fact that, in these conditions, drug solubility in hydrophilic fluids is usually much higher than that of the crystalline form,10,11 so that the drug dissolution rate is considerably increased.5 Because both nanocrystals and the amorphous drug are not stable and they tend to crystallize into the more thermodynamically stable macrocrystal size, the polymer, incorporating the drug inside its three-dimensional network, serves as a stabilizing agent hindering the macrocrystals formation. This action is due to the chemical and physical drug/polymer interactions and to the physical presence of the polymeric chains because macrocrystals can form on the condition that network meshes be sufficiently wide.5 Among the available techniques used for drug loading into cross-linked polymers, cogrinding has the considerable advantage of not requiring the use of solvents * To whom correspondence should be addressed. Tel.: 0039 040 8992423. Fax: 0039 040 8992243. E-mail: mariog@ dicamp.univ.trieste.it. † Eurand International SpA. ‡ DICAMP.

whose elimination from the final formulation can often represent a very expensive and delicate stage. Indeed, solvent must be eliminated without extracting the drug and without modifying the drug distribution inside the polymer because it is well-known that this distribution can sensibly affect the release kinetics.12,13 Polymer cogrinding, although commonly used since 1988,14 is a relatively new technique for the pharmaceutical field,15 and, to our knowledge, only one example exists of the industrial production of pharmaceutical systems based on this technology.16 A vibrational mill seems to be the most promising tool for cogrinding because of the high energy involved in the grinding process, the continuous character of energy transfer to materials (polymer and drug), and its easy scale-up to an industrial scale. Essentially, this mill realizes an energy transfer to polymer and drug because this mixture remains trapped in the collision zones among the grinding media (usually made by dense and high-abrasion-resistant materials such as alumina) and among the grinding media and the mill bowl.17,18 The energy supplied during each impact determines, in principle, polymer particle size reduction, drug crystal size reduction (the extreme consequence of this phenomenon leads to the amorphous drug), chemical reactions, phase transformation,19 drug loading into crosslinked polymeric particles,9 and, obviously, heat dissipation.20-22 This last aspect (heat dissipation) may play an important role in the cogrinding of pharmaceutical substances because many of them (especially therapeutic agents) are thermolabile. Consequently, the mill bowl temperature must not exceed a determined threshold for the therapeutic effectiveness and reliability of the cogrounded mixture. Accordingly, the aim of this work is to develop an analysis of heat flow inside a typical vibrational mill by means of the comparison between a proper mathematical model and experimental data. 2. Modeling A typical high-energy vibrational mill is made up of a bowl having, approximately, a toroidal shape and containing the grinding media (small high-density and

10.1021/ie020537g CCC: $25.00 © 2003 American Chemical Society Published on Web 03/26/2003

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smaller displacements (these displacements reduce as the mill load increases), we can reasonably suppose that the volume occupied by mGC does not appreciably increase during milling. The external chamber (EC) corresponds to the volume delimited by the torus, the mill lid chamber (ML), and the engine, represented by an additional chamber (EL). One of the main hypotheses on which the model relies is that, in each chamber, the temperature is supposed to be homogeneous and dependent only on time t. Finally, we suppose that the GC heating, because of grinding media collisions, is due to an internal dissipated power directly proportional to the engine power W. As a consequence, our model reads

mGCcpGC Figure 1. Schematic diagram of the mill, with external chamber EC, internal chamber IC, grinding chamber GC, mill lid ML, bowl wall W, EC height LEC, distance from the xx axis to the mill bowl center R, and external and internal radii of the mill re and ri. The arrows show the heat flow in the mill (asterisks represent grinding media).

Wη - hc1A1(TGC - TIC) - kpAb1

torus radius internal bowl radius external bowl radius EC height engine chamber thickness

0.190 0.089 0.117 0.179 0.007

high-abrasion-resistant cylinders). The upper part of the bowl is cut and covered by the mill lid, while the lower part is connected by springs to get a resiliently supported rigid body to a metallic base fixed to the ground. The bowl is rigidly connected to an electrical engine (placed approximately on the bowl symmetry axis) determining the motion of a shaft carrying, in its upper and lower parts, two eccentrics. When the engine runs, the rotation of the shaft provokes the motion of the two eccentrics, which, in turn, give origin to a torque determining the complex dynamics of the bowl/engine system. As a consequence, the movement of the grinding media inside the bowl produces the compound milling and the heat flow caused by the huge number of collisions. Because of the complexity of this system from the thermal analysis viewpoint, we are obliged to simplify the physical frame by introducing some geometrical and physical hypotheses. We assume that the real bowl (of course, it does not have a perfect toroidal shape) can be represented by a torus of revolution characterized by the same revolution (R) and internal radius (ri), while the external radius (re) is adjusted in order to confer the same volume to the real and approximated bowls (see Figure 1). Geometrical characteristics of the mill can be seen in Table 1. Additionally, the real mill is schematized as shown in Figure 1: the internal torus volume is subdivided into the grinding chamber (GC), corresponding to the volume occupied by the grinding media, and the internal chamber (IC), corresponding to the grinding media free volume. The hypothesis of a neat separation between the GC and IC is supported by an analysis (performed with Visual Nastran software) of the grinding media displacement when the mill runs. We could verify that the displacement of the outer milling media is approximately 27% of the GC thickness when the mill load mGC is 36 kg. Moreover, because the majority of the milling media (90%), lying in the inner part of the GC, undergoes much

|

∂Tb ∂r

r)ri

(1)

mICcpa

dTIC ) hc1A1(TGC - TIC) - hc2Ab2(TIC dt TbIC) - hc4iA2(TIC - TML) (2)

mECcpa

dTEC ) hc3Ab3(TEC - TbEC) + hc6A4(TEL dt TEC) - hc5A3(TEC - TML) (3)

mMLcps

dTML ) hc4iA2(TIC - TML) + hc5A3(TEC dt TML) - hc4e(A2 + A3)(TML - Ta) (4)

mELcps

dTEL ) Wζ - hc6A4(TEL - TEC) - hc7A5 dt (TEL - Ta) - hc7A6(TEL - Ta) (5)

Table 1. Geometrical Characteristics of the Mill (See Figure 1) R (m) ri (m) re (m) LEC (m) SEC (m)

dTGC ) dt

where mXY indicates the mass of the XY chamber, cpx is the specific heat at constant pressure of material X, TXY is the temperature of chamber XY, t is time, W is the electrical engine power, η is the fraction of the nominal engine power dissipated in the GC due to grinding media collisions, hcx is the heat-transfer coefficient, Ai and Abi are heat-transfer areas, Tb is the temperature of the generic position inside the mill bowl while TbIC and TbEC are the bowl temperatures on the surfaces faced on the IC and EC, respectively, Ta is the environmental temperature, r is the radial coordinate (ri < r < re), ζ is the fraction of the engine power W lost due to internal friction, and kp is the thermal conductivity of the mill bowl. Equation 1, the core of our model, expresses the heat balance evaluated for the GC, whose volume depends, obviously, on the mill load considered (although mGC expresses the mass of grinding media, the volume occupied by this mass accounts for the void fraction characterizing the random organization of grinding media poured inside the mill bowl). The first right-handside term represents the thermal power dissipated in the GC due to collisions, and it is the main term responsible for the temperature increase of the whole system. The second right-hand-side term indicates the thermal flow exchanged between the GC and IC, supposing that the separating area A1 does not vary because of mill motion and that energy transport occurs according to a natural convection mechanism. In principle, the

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Figure 2. Control volume shape and coordinate system. Figure 4. Critical angular positions for setting up the boundary conditions of eq 6.

balance evaluated on the generic control volume, neglecting the tangential flow, reads

Fpcpp

Figure 3. Each control volume generated by the rotation of a planar curve d about the xx axis. Areas and volumes of the control volume are calculated through Pappus-Guldin’s theorems.

heat-transfer resistance between grinding media and the mill wall should be considered, but because of the practical impossibility of its estimation (it depends on the pressure in the wall/grinding medium collision zone23), we renounce to consider it. This approximation, of course, will be embodied in the fitting parameters. Equations 2-4 represent respectively the heat balance performed on the IC and EC and in the ML. Because of the high thermal conductivity of the mill lid made of stainless steel, we suppose that its temperature is uniform despite the fact that it is faced to environments characterized by different temperatures (IC, EC, and external environment). Moreover, the small contact area involved makes the heat flow between the mill lid and the mill bowl negligible. Finally, eq 5 accounts for the heat balance in the engine chamber under the hypothesis that, because of internal friction, part of the engine power W is lost by contributing to the increase in the temperature. Because of the fact that the mill bowl is made up by a very low thermal conductivity material (polyurethane), the hypothesis of uniform temperature inside it cannot clearly hold. Accordingly, we suppose that the bowl temperature Tb depends on position, namely, on the radial (r) and angular (γ) coordinates, assuming, because of mill symmetry, that no thermal flux can occur in the tangential direction (ψ) (see Figure 2). Obviously, we assume that conduction is the only mechanism of heat transport inside the bowl, and that is why a partial differential equation is required to determine the temperature surface Tb(r,γ). Nevertheless, the particular shape of the mill bowl (torus of revolution) requires some comments about this equation. Its resolution requires subdivision of the mill bowl into small control volumes generated by the rotation of a planar surface about the mill symmetry axis xx (see Figure 3). Their total area and volume can be calculated through the first and second theorem of Pappus-Guldin.24 The heat

[

(

]

)

D0 ∂2Tb ∂Tb kp ∂ ∂Tb ) rD1 + ∂t rG ∂r ∂r r ∂γ2

(6)

where Tb(r,γ) is the temperature, Fp and cpp are respectively the polyurethane density and specific heat at constant pressure, G is the distance between axis xx and the centroid of the control volume cross-sectional surface (see Figure 3), D1 is the distance between the xx axis and the centroid of curve r dγ (this is the height of the control volume; see Figure 3), and D0 is the distance between the xx axis and the centroid of curve dr (this is the control volume thickness; see Figure 3). Equations 1-6 are solved with the initial condition that all chambers are at room temperature Ta. Moreover, eq 6 must accomplish the following boundary conditions (see Figure 4):

Tb|r)ri ) TGC

|

∂Tb ∂γ

|

∂Tb ∂r

) γ)γ0

|

∂Tb ∂γ

γ)γ4

γ1 e γ e γ3 )0

ri e r e re

(7) (8)

hc2(Tri,γ - TIC)

)

kp

r)ri

γ0 e γ e γ1

|

)

|

)

∂Tb ∂r

hc3(TEC - Tre,γ)

r)re

∂Tb ∂r

r)re

kp hca(Ta - Tre,γ) kp

γ3 e γ e γ4 (9) γ0 e γ e γ2 (10)

γ2 e γ e γ4

(11)

where hca is the heat-transfer coefficient between the bowl and environment. Equation 7 derives from the assumption of negligible thermal resistance between the GC and mill wall. The high difficulty of defining the control volume geometry in γ0 and γ4 (see Figure 4), and the negligible volume of these zones, allows us to approximate them as two adiabatic surfaces (see eq 8). Conditions (9)-(11) are based on the hypothesis that the heat flow occurs according to a natural convection mechanism. Undoubtedly, this choice could appear arbitrary because we are dealing with heat flow from moving bodies (mill bowl and lid and grinding media). This means that the phenomenology is very complex because both natural

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and forced convection mechanisms contribute to heat transfer. Simple calculations can be performed in the cases of mill lid and bowl, leading to similar values of the heat-transfer coefficients related to forced25 or natural26 convection mechanisms (if the maximum velocity experienced by the mill lid and bowl is set equal to 0.7 m/s, according to Visual Nastran dynamic simulation and experimental results obtained from an accelerometer). In both cases the estimation of the Reynolds and Grashof numbers, respectively, clearly indicate laminar regimes. However, the two contributions cannot be easily combined in the case under investigation, and, to our knowledge, an appropriate and reliable theoretical background does not exist to address such an operation. Indeed, bodies involved in the heat-transfer process undergo a pulsating motion, passing through quite different velocity regimes, and then the relative importance of natural and forced convection changes with time and should be measured by time-dependent heat-transfer coefficients. Even more complex appears the analysis of grinding media where proper relations for the estimation of the heat-transfer coefficients are lacking. On the basis of these considerations, a simplified analysis is compulsory for our purposes and natural convection must be considered an acceptable approximation and a convenient reference for the following numerical calculations. Because of the nature of the equations constituting our model, a numeric solution is required. Although, in principle, more sophisticated implicit techniques could be considered, eqs 1-5 are solved by means of a fifthorder Runge-Kutta method with an adaptive step size.27 Indeed, the physical conditions of our thermal problem are not those typically associated with a stiff system where the coexistence of very fast and very slow subphenomena takes place. Moreover, an a posteriori check on the model solution proved the suitability of the fifth-order adaptive step-size Runge-Kutta method because we did not meet any stability problem. Equation 6 is solved by the control volume method.28 The solution strategy requires that, at each time step t, eqs 1-5 are solved on the basis of the old temperatures [Ttb(r,γ), TtGC, TtIC, TtEC, TtEL, and TtML] to get the new values and, subsequently, eq 6 is solved on the basis t+dt t+dt of the new chamber temperatures (Tt+dt GC , TIC , TEC , t+dt t+dt TEL , and TML ) to yield the new temperature distribu(r,γ)]. Because of the tion inside the mill bowl [Tt+dt b variability of the GC volume (it depends on the mill load considered), we preferred to set eight critical control volumes in the angular direction, γ, accounting for the boundary condition discontinuity in γ0, γ1, γ2, γ3, and γ4 (see Figure 4 and eqs 7-11), allowing the number of control volumes to vary between them. To ensure solution accuracy, we chose 4 control volumes in the radial direction and 24 (8 + 16) in the angular direction for a total of 96 control volumes. In light of the fact that the system of ordinary differential equations (1)-(5) has to be coupled with the partial derivative differential equation (6) and with the aim of obtaining a good accuracy of the numerical solution, we decided to set a very small initial time step dt ()1 s). 3. Materials and Method The heat-transfer coefficients are estimated by means of the following correlation:25,26

hc ) Cka(GrLPr)1/4/L

(12)

Table 2. Heat-Transfer Coefficient Constants C (Equation 12) and Physical Properties of the Materials Constituting the Mill (k ) Thermal Conductivity; cp ) Specific Heat at Constant Pressure; G ) Density) heat-transfer coefficient constant hc2 hc3 hc4i hc4e hc5 hc6 hc7

C ) 0.47 C ) 0.55 C ) 0.27 C ) 0.54 C ) 0.27 C ) 0.54 C ) 0.55

air

grinding media

ka (W/m‚K) ) 0.027 cpa (J/K‚kg) ) 1005 Fa (kg/m3) ) 1.14 polyurethane kp (W/m‚K) ) 0.17 cpp (J/K‚kg) ) 1820 Fp (kg/m3) ) 1155

FGC (kg/m3) ) 2100 cpGC (J/K‚kg) ) 921.1 steel Fs (kg/m3) ) 7820 cps (J/K‚kg) ) 460.6

where C values are expressed in Table 2, ka is the thermal conductivity of the air, Gr and Pr are the Grashof and Prandtl numbers, and L is the characteristic length. Polyurethane thermal conductivity was determined in the CIMAC laboratory (Vigevano, Pavia, Italy) according to the Norme Franc¸ aise method,29 and its value is kp ) 0.17 W/K‚m, while the polyurethane specific heat at constant pressure was measured by differential scanning calorimetry (Perkin Helmer Corp., Norwalk, CT), and its value is cpp ) 1820 J/K‚kg. All other material physical properties were set according to the literature25,26,30 and can be seen in Table 2. The GC density was determined by the measure of the mass of an assigned volume, and its value is FGC ) 2100 kg/m3. To simplify our analysis, we decided not to consider the thermal effect of grinding materials (namely, drug and polymeric carriers) because their masses are negligible in comparison to that of the grinding media. Accordingly, experimental tests are carried out on loading of the GC with the grinding media only. In particular, five different mill load conditions were explored (36, 39, 43, 49, and 55 kg) because this is the range in which the mill was designed to work. A frequency inverter (Siemens Micromaster, Munich, Germany) ensures that each experimental test is led at constant engine revolution speed (1100 rpm) while temperatures are measured by an in situ thermocouple device (Futura Temperature Tester 177; accuracy ( 0.2 K; IC and environmental temperatures) and by a portable infrared thermometer (Kane-May KANST2PG, Hertfordshire, U.K.; accuracy ( 0.5 K; grinding media, engine cover, and bowl wall). Each experiment was duplicated. 4. Results and Discussion In light of our assumptions, the model has only three fitting parameters: the fraction of power lost as a result of internal engine friction (ζ; this is equal to 1 engine efficiency), the heat-transfer coefficients relative to the heat transfer between GC and IC (hc1), and the fraction of the nominal engine power W dissipated within the GC as a result of grinding media collisions (η). Although we should determine these three parameters by means of a simultaneous model fitting on the experimental data referring to GC, IC, and engine chamber, we adopted a different and simpler strategy. Because of the thickness (28 mm) and the good thermal insulation of the polyurethane bowl, the temperatures of both the GC and IC are mainly dependent on η and hc1 (see eqs 1 and 2), and they undergo a very small influence from ζ. Therefore, ζ is determined by setting whatever reasonable η and hc1 values and fitting the model on the temperature evolution that takes place within the

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Figure 5. Time evolution of the engine temperature TEL (mGC ) 43 kg). The solid line represents the model best fitting, and open circles represent experimental points.

Figure 6. Time evolution of the GC and IC temperatures, TGC and TIC, respectively (mGC ) 43 kg). The solid and dashed lines represent the model best fitting (solid line, GC; dashed line, IC). Open and solid circles represent experimental data (GC, open circles; IC, solid circles).

engine chamber. Then, on the basis of this ζ value, the model is simultaneously fitted on the data referring to the IC and GC. Figure 5 shows the reasonably good agreement between the model best fitting (solid line) and experimental data (open circles) of the engine chamber temperature working with mGC ) 43 kg of mill load. The fitting procedure yields ζ ) 0.027, which is in good agreement with the value of 0.03 found in the literature.31 Interestingly, by means of the same ζ value, the model is able to fit well the other conditions explored (36 and 55 kg; they are similar to data shown in Figure 5), underlying the fact that this parameter is mill load independent. Figure 6 reports the comparison between the experimental data (GC, empty circles; IC, black circles) and the model best fitting (solid line, GC; dashed line, IC) in the mGC ) 43 kg case when assuming ζ ) 0.027. Because the data are reasonably well described, we can say that the adopted model takes properly into account the most important phenomena affecting the temperature rise in the GC and IC, respectively. The fitting procedure yields hc1 ) 3.4 W/K‚m2 and η ) 0.31. In the same manner, Figures 7 and 8 show comparisons between the model best fitting (solid line, GC; dashed line, IC) and the experimental data (GC, empty circles; IC, black circles) in the mGC ) 36 kg and mGC ) 55 kg cases, respectively. Again the model is able to fit reasonably well the experimental data (being always ζ ) 0.027) in both cases but assuming different values for the fitting parameters. In particular, we have hc1 ) 6.7 W/K‚m2 and η ) 0.22 for mGC ) 36 kg and hc1 ) 2.4 W/K‚m2 and η ) 0.47 for mGC ) 55 kg. To generalize

Figure 7. Time evolution of the GC and IC temperatures, TGC and TIC, respectively (mGC ) 36 kg). The solid and dashed lines represent the model best fitting (solid line, GC; dashed line, IC). Open and solid circles represent experimental data (GC, open circles; IC, solid circles).

Figure 8. Time evolution of the GC and IC temperatures, TGC and TIC, respectively (mGC ) 55 kg). The solid and dashed lines represent the model best fitting (solid line, GC; dashed line, IC). Open and solid circles represent experimental data (GC, open circles; IC, solid circles).

our results, it is convenient to propose two empirical equations able to describe the hc1 and η dependence on mGC. In particular, we found that hc1 depends on mGC according to the following empirical equation:

(

hc1 ) hc0 a +

b c + mGC m

2

GC

)

(13)

where a ) 11.45, b ) -1081.63 kg, and c ) 28477 kg2 and hc0 is calculated according to the following equation (natural convection in laminar flow25,26):

hc0 ) 0.54ka(GrLPr)1/4/L

(14)

For η, the following linear expression applies:

η ) 0.0099mGC - 0.0952

(15)

The experimental behavior of hc1 (reduction with mGC) and η (increase with mGC) can be explained in light of the grinding media dynamics during the milling process. On the basis of experimental observations and theoretical simulations led by a proper software (Visual Nastran), we can say that when mGC decreases, a higher kinetic energy is transferred to the grinding media, which experience a wider displacement in their movement. This determines a more efficient heat transfer and a larger interface area between GC and IC. Because, for the sake of simplicity, we suppose the heattransfer area, A1, is independent of mill movement, our model interprets the above-described situation by means

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Figure 9. Time evolution of the GC and IC temperatures, TGC and TIC, respectively (mGC ) 39 kg). The solid and dashed lines represent the model prediction (solid line, GC; dashed line, IC). Open and solid circles represent experimental data (GC, open circles; IC, solid circles).

ingly, the physical and geometrical hypotheses, on which the model was built, are reasonable also because the model can predict correctly the temporal evolution of the mill temperature (GC and IC temperature) for two mill loads different from those used for the fitting. In particular, the simple idea of subdividing the mill into different chambers, characterized by a uniform spatial temperature, turned out to be reliable, provided that heat transfer in the mill bowl is described in terms of a two-dimensional conduction problem. For this purpose, it is worth mentioning that the numerical solution adopted obliges to define a particular shape for the control volume that is not usually employed.27,32-34 Indeed, the toroidal shape of the mill bowl avoids the adoption of the common (cylinders, spheres, and parallelepipeds) control volumes. Moreover, this study demonstrates that mill load can be used as an experimental parameter to control the temperature rise in the mill bowl. In particular, the bigger the mill load, the more pronounced the temperature rise is. List of Symbols

Figure 10. Time evolution of the GC and IC temperatures, TGC and TIC, respectively (mGC ) 49 kg). The solid and dashed lines represent the model prediction (solid line, GC; dashed line, IC). Open and solid circles represent experimental data (GC, open circles; IC, solid circles).

of a hc1 increase upon a mGC decrease. The η dependence on mGC can be explained by remembering that energy dissipation in the GC depends on the number of impacts per unit time and on the energy involved in each impact. These two phenomena are, in turn, directly and inversely proportional to mGC as explained above. Accordingly, because an η increase with mGC is observed, we have to conclude that the huge increase of impact as a result of the mGC increase not only compensates for the reduced energy involved in each impact but also determines an increase of the total energy dissipated. To further check the reliability of the developed model, we make a comparison between the experimental data and model predictions by assuming two different mill loads (mGC ) 39 and 49 kg), hc1 and η calculated according to eqs 13 and 15. Figures 9 and 10 show that the predictive ability of the model is satisfactory in both cases for what concerns the GC and IC temperatures TGC and TIC. Interestingly, an inspection of Figures 6-10 reveals that the reduction of mill load determines a smaller temperature increase. Accordingly, mill load is an important parameter in the control of the temperature rise in the mill bowl. 5. Conclusions Because of the fact that the proposed model is able to fit in a satisfactory manner the experimental dynamic evolution of the temperature within the GC, IC, and engine chamber, we can conclude that it is reliable for the thermal description of our vibrational mill. Accord-

A1 ) heat-transfer area between the grinding and internal chambers (m2) A2 ) area of the mill lid in contact with the internal chamber (m2) A3 ) area of the mill lid in contact with the external chamber (m2) A4 ) area of the engine cover lid (m2) A5 ) lateral side area of the engine cover (m2) A6 ) bottom area of the engine cover (m2) Ab1 ) bowl area in contact with the grinding chamber (m2) Ab2 ) bowl area in contact with the internal chamber (m2) Ab3 ) bowl area in contact with the external chamber (m2) cpa ) specific heat at constant pressure of air (J/K‚kg) cpb ) specific heat at constant pressure of the ball (J/K‚ kg) cpp ) specific heat at constant pressure of polyurethane (J/K‚kg) cps ) specific heat at constant pressure of steel (J/K‚kg) D1 ) distance from the xx axis to the centroid of curve r dγ (m) D0 ) distance from the xx axis to the centroid of curve dr, placed in the bottom face of the control volume (m) EC ) external chamber G ) distance from the xx axis to the centroid of the planar curve that generates the subvolume (m) Gr ) Grashof number IC ) internal chamber hc1 ) heat-transfer coefficient between the grinding and internal chambers (J/K‚s‚m2) hc2 ) heat-transfer coefficient between the bowl and internal chamber (J/K‚s‚m2) hc3 ) heat-transfer coefficient between the bowl and external chamber (J/K‚s‚m2) hc4e ) heat-transfer coefficient between the mill lid and ambient (J/K‚s‚m2) hc4i ) heat-transfer coefficient between the internal chamber and mill lid (J/K‚s‚m2) hc5 ) heat-transfer coefficient between the external chamber and mill lid (J/K‚s‚m2) hc6 ) heat-transfer coefficient between the engine cover lid and external chamber (J/K‚s‚m2) hc7 ) heat-transfer coefficient between the lateral side of the engine cover and ambient (J/K‚s‚m2) hc8 ) heat-transfer coefficient between the bottom of the engine cover and ambient (J/K‚s‚m2) kp ) thermal conductivity of polyurethane (J/K‚s‚m)

Ind. Eng. Chem. Res., Vol. 42, No. 9, 2003 2021 maE ) mass of the air in the external chamber (kg) maI ) mass of the air in the internal chamber (kg) mEnC ) mass of the engine cover (kg) mGC ) mass of the grinding media (kg) mML ) mass of the mill lid (kg) L ) characteristic length (m) LEC ) external chamber height (m) Pr ) Prandtl number ri ) internal radius of the mill bowl (m) re ) external radius of the mill bowl (m) r ) radius of the mill bowl (ri < r < re) (m) R ) distance from the xx axis to the mill bowl center (m) Ta ) ambient temperature (K) TEC ) external chamber temperature (K) TEL ) engine cover temperature (K) TGC ) grinding chamber temperature (K) TIC ) internal chamber temperature (K) TML ) mill lid temperature (K) T ) temperature (K) t ) time (s) W ) electrical engine power (J/s) Greek Symbols ζ ) fraction of electrical energy lost by the engine because of internal friction η ) fraction of engine power supplied to the grinding chamber Fp ) polyurethane density (kg/m3)

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Received for review July 22, 2002 Revised manuscript received February 14, 2003 Accepted February 17, 2003 IE020537G